Relativity time dilation

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Special Relativity and Time Dilation: Einstein’s Calculation Errors

Alasdair N. Beal BSc CEng FICE FIStructE

anbeal@btinternet.com

Physics Essays 38, 2 (2025) pp. 112, 113

Abstract

In 1905 Einstein presented his Special Theory of Relativity and derived transformation equations from it, which he then used to calculate time dilation as τ = t√(1-v²/c²). However his calculation included inconsistent assumptions and failed to consider relevant equations from his relativity analysis. When these errors are corrected, the correct time dilation equation based on his theory is τ = t√(1-v²/c²)/(1+v/c).

Therefore if Einstein’s theoretical explanation for time dilation is correct, his equation for it is wrong and slowing of a moving clock should be greater than he claimed. Alternatively, if his equation for time dilation is correct, then his theoretical explanation for it must be wrong and a different explanation is required.

Keywords: Einstein; Special Relativity; Time Dilation; Clocks Paradox; Mathematical Errors; Transformation Equations; Lorentz Equations.

Introduction

In 1905, Einstein presented his Special Theory of Relativity and derived transformation equations for time and distance from it which match those of the Lorentz Transformation [1]:

τ = (t - vx/c²)/√(1-v²/c²) (1)

ξ = (x - vt)/√(1-v²/c²) (2)

He then calculated time dilation from these equations by considering a clock at the origin of coordinate system k, which moves at velocity v relative to ‘stationary’ system K. By inserting the X-coordinate of this clock (x = vt) into equation (1) he obtained:

τ = t√(1-v²/c²) (3)

From equation (3) he calculated that, when viewed by a stationary observer, a clock moving at velocity v should appear to run slow by approximately ½v²/c² seconds per second.

(Note: for ease of cross-reference Einstein’s original notation is used here: x, y, z are coordinates in ‘stationary system’ K, ξ, η, ζ are coordinates in ‘moving system’ k; and t and τ are times shown by clocks synchronised in systems K and k respectively.)

Analysis

In his 1905 paper Einstein did not define the meanings of the terms x and t in his transformation equations (1) and (2). However if he derived these equations from his relativity calculations, where x and t have specific, particular meanings, then for mathematical consistency they should have the same meanings in his subsequent calculations.

His relativity analysis considers events in two coordinate systems: ‘stationary’ system K and ‘moving’ system k, which travels at velocity v relative to K. For points on the X-axis, a light flash emitted from the origin of system k travels to a remote point at coordinate ξ and arrives there at time τ; in system K it travels to coordinate x and arrives there at time t. Einstein’s calculations include the following assumptions: (i) at t = 0, τ = 0, (ii) at t = 0 the origins of systems k and K coincide and (iii) at t = 0 the system K coordinate of the remote point is x'.

In this analysis ξ and x are not independent variables: they are functions of τ and t respectively, as defined by Einstein in the following equations:

x' = x - vt (4)

x'/(c-v) = t (5)

ξ = cτ (6)

Einstein calculates time dilation by considering a clock which is at the origin of ‘moving’ system k, so its coordinates are x' = 0, ξ = 0 and y = z = 0. If we ignore equations (5) and (6) and simply insert x' = 0 into equation (4), then x = vt, as assumed by Einstein. However if x' = 0 is inserted in equation (5), then t = 0 and if ξ = 0 is inserted in equation (6), then τ = 0. Furthermore, if equation (4) x' = x - vt is combined with equation (5) x' = ct -vt, then x = ct. However if x = ct and x = vt are both true then the conclusion must again be t = 0.

Thus Einstein’s time dilation calculation ignores relevant equations and is not consistent with the assumptions of his relativity analysis. If all of his equations (4) - (6) and the assumptions of his relativity analysis are taken into account, then if a clock is at the origin of system k the only valid conclusion that can be reached is: t = 0 and τ = 0. The reason for this is simple: in Einstein’s thought experiment, if the clock is at the system k origin then the distance travelled by the light flash will be zero and his analysis cannot tell us anything useful about its timekeeping.

Einstein’s relativity analysis can only produce meaningful conclusions if the clock is not at the origin of system k (i.e. x' ≠ 0 and ξ ≠ 0). In this situation, according to his equations (4) - (6) x = ct and ξ = cτ, so his transformation equations (1) and (2) become:

τ = t(1 - v/c)/√(1-v²/c²) (7)

ξ = x(1 - v/c)/√(1-v²/c²) (8)

Alternatively, equation (7) can be expressed as:

τ = t√(1-v²/c²)/(1+v/c) (9)

Thus if the assumptions of Einstein’s relativity analysis are applied consistently throughout the calculation, then his equation (3) for time dilation is wrong: the correct solution is equation (9). The following table compares the clock slowing predicted by these equations for various values of v/c. As can be seen, equation (9) predicts much greater clock slowing than equation (3), particularly for lower values of v/c (Table I).

Table I
Comparison of slowing of moving clock (sec/sec) calculated from Eq. (3) and (9)

v/c Eq. (3) clock slowing Eq. (9) clock slowing

0.01 0.00005 0.010

0.1 0.005 0.095

0.2 0.02 0.184

0.3 0.046 0.266

0.4 0.083 0.345

0.5 0.134 0.423

Conclusions

Einstein’s calculation of time dilation for a moving clock at the origin of ‘moving’ system k is based on inconsistent assumptions and ignores relevant equations from his relativity analysis. When these errors are corrected, his analysis leads to only one possible conclusion: t = 0 and τ = 0 and therefore his equation τ = t√(1-v²/c²) is incorrect.

Einstein’s relativity analysis can only produce meaningful conclusions if the ‘moving’ clock is not at the origin of system k. In this situation x = ct and ξ = cτ, so his transformation equations become:

τ = t(1 - v/c)/√(1-v²/c²) (7)

ξ = x(1 - v/c)/√(1-v²/c²) (8)

Thus, based on his theory, the equation for time dilation should be:

τ = t√(1-v²/c²)/(1+v/c) (9)

Therefore:

(a) if the explanation for time dilation which Einstein presented in 1905 is correct, then his equation (3) is wrong - the correct solution is equation (9), so the apparent slowing of a moving clock should be much greater than he predicted; or

(b) alternatively, if Einstein’s equation (3) is correct, then his theoretical explanation for time dilation must be wrong and a different explanation is required.

These conclusions have important consequences for the interpretation of experiments designed to test Einstein’s Special Theory of Relativity by measuring apparent time dilation.

References

1. A. Einstein, Ann. Phys. 17, 891 (1905). H. A. Lorentz, A. Einstein, H. Minkowski & W. Weyl, The Principle of Relativity, trans. W. Perrett & G. B. Jeffery (Methuen, London, 1923/ Dover reprint, Mineola, NY, 1952).

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